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Chapter 12: Problem 22

An organ pipe that is open at both ends has a fundamental frequency of $382 \mathrm{Hz}\( at \)0.0^{\circ} \mathrm{C}$. What is the fundamental frequency for this pipe at \(20.0^{\circ} \mathrm{C} ?\)

Short Answer

Expert verified
Answer: The fundamental frequency of the open-ended organ pipe at 20 degrees Celsius is approximately 396 Hz.

Step by step solution

01

Find the speed of sound at 0 degrees Celsius

First, we need to find the speed of sound in the air at 0 degrees Celsius. We can use the following formula to find the speed of sound in the air: \(v = 331.4 \sqrt{1+\frac{T}{273.15}}\) Where: \(v\) is the speed of sound in meters per second (m/s) \(T\) is the temperature in Celsius For this problem, \(T = 0.0^{\circ} \mathrm{C}\), so we plug this into the equation: \(v = 331.4 \sqrt{1+\frac{0}{273.15}} \approx 331.4 \, \mathrm{m/s}\)
02

Calculate the wavelength of the fundamental frequency at 0 degrees Celsius

Next, we will find the wavelength of the fundamental frequency at 0 degrees Celsius. To find the wavelength of the fundamental frequency, we will use the following equation: \(\lambda = \frac{v}{f}\) Where: \(\lambda\) is the wavelength in meters (m) \(v\) is the speed of sound in meters per second (m/s) \(f\) is the frequency in Hertz (Hz) For this problem, the frequency at 0 degrees Celsius is 382 Hz. So, we plug this into the equation: \(\lambda = \frac{331.4 \, \mathrm{m/s}}{382 \, \mathrm{Hz}} \approx 0.867 \, \mathrm{m}\)
03

Find the speed of sound at 20 degrees Celsius

Now, we need to find the speed of sound in the air at 20 degrees Celsius. We use the same formula as in step 1 with \(T = 20.0^{\circ} \mathrm{C}\): \(v = 331.4 \sqrt{1+\frac{20}{273.15}} \approx 343.4 \, \mathrm{m/s}\)
04

Calculate the fundamental frequency at 20 degrees Celsius

Finally, we can now find the fundamental frequency at 20 degrees Celsius using the new speed of sound and the same wavelength as calculated in step 2: \(f = \frac{v}{\lambda}\) Plugging in the values we have: \(f = \frac{343.4 \, \mathrm{m/s}}{0.867 \, \mathrm{m}} \approx 396 \, \mathrm{Hz}\) So, the fundamental frequency for the organ pipe at \(20.0^{\circ} \mathrm{C}\) is approximately \(396 \, \mathrm{Hz}\).

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Most popular questions from this chapter

A cello string has a fundamental frequency of \(65.40 \mathrm{Hz}\) What beat frequency is heard when this cello string is bowed at the same time as a violin string with frequency of \(196.0 \mathrm{Hz} ?\) [Hint: The beats occur between the third harmonic of the cello string and the fundamental of the violin. \(]\)
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A sound wave arriving at your ear is transferred to the fluid in the cochlea. If the intensity in the fluid is 0.80 times that in air and the frequency is the same as for the wave in air, what will be the ratio of the pressure amplitude of the wave in air to that in the fluid? Approximate the fluid as having the same values of density and speed of sound as water.
Blood flow rates can be found by measuring the Doppler shift in frequency of ultrasound reflected by red blood cells (known as angiodynography). If the speed of the red blood cells is \(v,\) the speed of sound in blood is \(u,\) the ultrasound source emits waves of frequency \(f\), and we assume that the blood cells are moving directly toward the ultrasound source, show that the frequency \(f_{r}\) of reflected waves detected by the apparatus is given by $$f_{\mathrm{r}}=f \frac{1+v / u}{1-v / u}$$ [Hint: There are two Doppler shifts. A red blood cell first acts as a moving observer; then it acts as a moving source when it reradiates the reflected sound at the same frequency that it received.]
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